Analyzing Dynamical Systems
An Overview of Mathematical Methods for Analyzing Systems of Differential Equations
After formulating a system of differential equations, a next step to do some analysis and try to understand the model. Here, we present an overview of methods for analyzing systems of differential equations.
Anyone who has taken advanced undergraduate classes in dynamical systems will know something about the basic methods for qualitative analysis of differential equations. A prerequisite is a solid background in linear algebra, particularly eigenvalues and eigenvectors. This vignette takes a birds-eye view of the topic for malaria analysts.
We have no intention of covering the mathematics in detail. We do not believe it is important for malaria analysts to develop a deep understanding of the underlying mathematics, but we do believe that every malaria analyst should find some friends who do. We recommend the
Solving
In Solving Dynamical Systems (above), we explain what it means to solve a systems of differential equations, and we gave some examples.
Qualitative Analysis
Phase Plane
Stability Analysis
A basic question to ask about dynamical systems is whether there are any stable states, and whether those steady states are stable. By stable we mean that small perturbations away from the steady state tend to return to it. We cover this in the Stability Analysis
Malaria Analytics
In malaria analytics, we are not as interested in analyzing a model as we are in learning what that model can teach us about malaria.
Thresholds
Scaling Relationships
Suggested Reading
- Steven H. Strogatz, Nonlinear Dynamics and Chaos. Available online